Recent content by Threepwood

Homework Statement I have an equation for a unitary matrix U, \sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} - \tilde{\epsilon}_k \delta_{qm}} \right)} = 0 I need to solve this equation for U Homework Equations The property of...

Homework Statement I have an equation for a unitary matrix U, \sum_k{ \left(\left(\varepsilon_k - \mu\right) \bar{U}_{qk} U_{km} + \gamma \sum_p{\bar{U}_{qk}U_{pm} - \tilde{\varepsilon}_k \delta_{qm}} \right)} = 0 I need to solve this equation for U Homework Equations The property of...

That was never stated in the question, but maybe it was implied somehow. It would make sense. How would I go about doing that?

Ok, but what about finding an equation for U?

Yes, they are. At the moment I'm more interested in finding this equation for U, but I have no idea where to even start. I've just been playing around with the relations, like taking c_p c_q^{\dag} + c_q^{\dag} c_p = \delta_{pq} applying c_q to the left c_q c_p c_q^{\dag} + c_q c_q^{\dag} c_p =...

I need to prove those relations. How do I prove that \{b_q , b_p\} = 0 and \{b_q , b_p^{\dag} \} = \delta_{pq}? And also, beyond that, how do I find an equation for U? I don't need to solve the equation for U, just find it.

Isn't that precisely what I'm supposed to be proving?

Homework Statement I have been given the Hamiltonian H = \sum_{k}\left(\epsilon_k - \mu\right) c_k^{\dag} c_k + \gamma \sum_{kp}c_k^{\dag} c_p and also that c_p = \sum_{q} U_{pq} b_q I have to prove that this matrix U_{pq} is unitary, and find an equation for U_{pq}. Homework Equations...